Problem 4 · AMC 8 Stretch
Core
Logic & Word Problems
Arithmetic & Operations
work-backwardintelligent-guessing-and-testing
Jim asks a classmate how old she is. She answers, 'I was 14 the day before yesterday, but I'll be 17 next year.' On what day is she talking, and when is her birthday?
Show answer
Answer: She is speaking on January 1, and her birthday is December 31
Show hints
Hint 1 of 4
This sounds impossible, but it isn't! The secret is that a few days near New Year's can touch several different calendar years at once.
Still stuck? Show hint 2 →
Hint 2 of 4
If she was 14 just a couple of days ago but turns 17 'next year,' her age has to climb 14, 15, 16, 17. That's a lot of birthdays squeezed close together — so her birthday must sit right at the edge of the year.
Still stuck? Show hint 3 →
Hint 3 of 4
Try guessing her birthday is December 31. Then try having the conversation on January 1. Walk through the days: the day before yesterday, yesterday, today, the next December 31, and the one after that.
Show solution
Approach: Guess a year-edge birthday, then test the timeline
- This only seems like a paradox if you forget that 'next year' on the calendar can be just a day or two away. Let her birthday be December 31 and the conversation happen on January 1.
- The day before yesterday = December 30: she hadn't yet had her Dec 31 birthday, so she was 14 (matches 'I was 14 the day before yesterday').
- Yesterday = December 31: she turned 15. Today = January 1: she is 15.
- This coming December 31 she turns 16; the December 31 after that she turns 17. Since today is January 1, that birthday lands in the NEXT calendar year (matches 'I'll be 17 next year').
- Everything fits: she is speaking on January 1, and her birthday is December 31.
Mark:
· log in to save